30-60-90 Triangle - Sides, Formula, Examples

30-60-90 Triangle: In geometry, a triangle is a three-sided polygon and is of many types—like equilateral, isosceles, scalene, acute, and obtuse triangles. However, some triangles are identified as special triangles because their 30 60 90 triangle sides and angles always follow the same pattern. One such triangle is the 30-60-90 triangle.

So, "What is a 30-60-90 Triangle?" A 30 60 90 triangle is a special right-angled triangle that has angles of 30°, 60°, and 90° that are always in the ratio 1:2:3, and the sides also follow a fixed 30 60 90 triangle ratio. Because of this, the 30-60-90 triangle theorem and formulas are very helpful in solving math problems quickly and easily.

30 60 90 Triangle Sides Overview

In a 30-60-90 triangle, the lengths of the sides always follow a fixed pattern based on the angles. Let’s understand this with triangle ABC, where ∠C = 30°, ∠A = 60°, and ∠B = 90°. The 30 60 90 triangle properties make it one of the most important special right triangles in geometry. According to the 30 60 90 triangle rules, the angles are always 30°, 60°, and 90°, and the 30 60 90 triangle sides ratio is fixed as 1 : √3 : 2.

• The side AB, which is opposite to the 30° angle, is the shortest side. We can call it y.

• The side BC, which is opposite to the 60° angle, is longer than AB. It is always equal to y × √3 or y√3.

• The side AC, which is opposite to the 90° angle, is the longest side and is the hypotenuse. It is always 2y.

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30 60 90 Triangle Theorem

The 30 60 90 triangle theorem tells us about a special rule between the sides of a triangle that has angles of 30°, 60°, and 90°. This rule helps us find the missing side lengths easily if we know just one side.

Statement:

In a 30-60-90 triangle,

• The hypotenuse (opposite the 90° angle) is twice the length of the shortest side.

• The longer leg (opposite the 60° angle) is √3 times the shortest side.

So, if the shortest side is y (which is the side opposite to the 30° angle), then:

• Hypotenuse = 2y

• Other side = y√3

This is the main idea behind the 30 60 90 theorem formula, and it is very useful for solving geometry questions without needing complex calculations.

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What is the 30 60 90 Formula?

The 30 60 90 formula shows the fixed ratio of the sides in a triangle with angles of 30°, 60°, and 90°. This 30 60 90 triangle ratio helps us find the missing sides quickly, just by knowing one side.

In a 30-60-90 triangle:

• The side opposite the 30° angle (shortest side) is taken as 1.

• The side opposite the 60° angle (medium side) is √3.

• The side opposite the 90° angle (hypotenuse) is 2.

So, the 30-60-90 triangle formula is: 1 : √3 : 2 — this is the ratio of the three sides.

There is also a ratio for the angles: 1 : 2 : 3, which matches the angles of 30°, 60°, and 90° in the triangle. By using this fixed ratio, we can easily apply the 30 60 90 theorem formula while solving different math problems.

Read More: What is Hexagon?

30 60 90 Triangle Properties

The 30 60 90 triangle properties help in solving problems quickly without measuring. By remembering the 30 60 90 triangle rules, students can easily calculate missing sides, since the hypotenuse is always twice the shortest side. The 30 60 90 triangle sides ratio is therefore very useful in trigonometry and real-life applications like construction and design.

30-60-90 Triangle Theorem Proof

To understand the 30-60-90 triangle theorem, let’s take an equilateral triangle ABC where all sides are equal and each angle is 60°. Let the side length of triangle ABC be a.

Now, draw a perpendicular from point A to side BC, and let it meet BC at point D. Since ABC is an equilateral triangle, this perpendicular line will cut BC exactly in half. Now we have formed two right-angled triangles: ABD and ADC.

Let’s take the triangle ABD.

• AB = a (original side of the equilateral triangle)

• BD = a/2 (half of BC)

• ∠ADB = 90° (as we drew a perpendicular)

• ∠B = 60°, so ∠A = 30°.

Now triangle ABD is a 30-60-90 triangle. We will now use the Pythagoras theorem to find the height AD:

Now, the three sides of triangle ABD are:

• Shortest side (BD) = a/2

• Medium side (AD) = (a√3)/2

• Hypotenuse (AB) = a

So, the side lengths are in the ratio: a/2 : (a√3)/2 : a

If we simplify this ratio by multiplying all sides by 2 and dividing by a, like (2a)/(2a) : (2a√3)/(2a): (2a/a), we get: 1 : √3 : 2, which proves the 30-60-90 triangle theorem.

For example, in triangle PQR, if ∠R = 30°, ∠P = 60°, ∠Q = 90°, and the side opposite to the 30° angle (QR) is 5 cm, then what will be the lengths of the other two sides?

Solution: Using the 30 60 90 theorem formula:

• Shortest side (opposite 30°) = y = 5

• Hypotenuse (opposite 90°) = 2y = 2 × 5 = 10 cm

• Other side (opposite 60°) = y√3 = 5√3 ≈ 8.66 cm

So, the side lengths of triangle PQR are:
QR = 5 cm, PR ≈ 8.66 cm, and PQ = 10 cm.

Read More: Top 10 Tricks to Master Mental Maths

How to Find the Area of a 30-60-90 Triangle?

To find the area of any triangle, we use the basic formula: Area = (1/2) × base × height.

In a 30-60-90 triangle, since it is a right-angled triangle, the height is the side that is perpendicular to the base. For instance, in triangle ABC:

• ∠C = 30°, ∠A = 60°, ∠B = 90°

• The base BC = a (this is the side opposite to the 60° angle).

• The perpendicular height = side opposite to 30° = a/√3 (using the 30-60-90 triangle formula)

Now, using the formula:

Therefore, the area of a 30-60-90 triangle, when the base is given as a, is: Area = a² ÷ (2√3)

 30-60-90 Triangle Examples With Solutions

Example 1: A garden is shaped like a 30-60-90 triangle. The longest side (hypotenuse) of the garden is 24 meters. What is the perimeter of the garden?

Solution: In a 30-60-90 triangle, the sides have the ratio of: 1 : √3 : 2

Here, the hypotenuse = 24 m = 2y
So, y = 2/42 = 12m (this is the shortest side).

The side opposite the 60° angle = y × √3 = 12 × 1.732 ≈ 20.78 m

Now, the perimeter = sum of all sides
= shortest side + side opposite 60° + hypotenuse
= 12 + 20.78 + 24 = 56.78 meters

Example 2: A triangle has sides 4√2, 4√6, and 8. Find the angles of this triangle.

Solution:  Given sides are: 4√2, 4√6, and 8

First, check if these sides follow the 30-60-90 ratio (1 : √3 : 2).

Divide each side by 4√2:

• 4√2 ÷ 4√2 = 1

• 4√6 ÷ 4√2 = √6 ÷ √2 = √3

• 8 ÷ 4√2 = 8 ÷ (4√2) = 2 ÷ √2 = (2√2) ÷ 2 = √2

So, the sides are in the ratio: 1 : √3 : √2

Since the last ratio is √2, not 2, the sides do not follow the 30-60-90 triangle rule exactly.

Therefore, this triangle is not a 30-60-90 triangle.

Example 3: The shortest side of a 30-60-90 triangle is 7 cm. Find the length of the hypotenuse and the side opposite the 60° angle.

Solution: Shortest side = y = 7 cm

Hypotenuse = 2y = 2 × 7 = 14 cm

Side opposite 60° = y√3 = 7 × 1.732 = 12.12 cm

Example 4: In triangle XYZ, the side opposite the 60° angle is 9√3 cm. Find the lengths of the other two sides.

Solution: We know the sides of a 30-60-90 triangle are in the ratio: 1 : √3 : 2

Given the side opposite 60° = 9√3 cm = y × √3

To find the shortest side y, divide by √3:
y = (9√3) ÷ √3 = 9 cm

Now, find the hypotenuse:
Hypotenuse = 2y = 2 × 9 = 18 cm.

The shortest side is 9 cm, and the hypotenuse is 18 cm.

Also Read: Brackets in Maths: Types, Rules & Examples

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